Simulation of a macrosegregation benchmark in a cylindrical coordinate system with a meshless method

Abstract

Abstract The main objective of the present paper is to define a new benchmark test for macrosegregation in axisymmetry and to verify a novel meshless method on it. The test case represents a solidification of Al4.5\u202fwt%Cu alloy in two different types of geometries, a solid and a hollow cylinder, cooled at the vertical boundaries. The volume averaging method is used to formulate the coupled mass, energy, momentum, and species transport equations for solid-liquid flow. The lever rule is used for determination of liquid and solid fraction. The meshless numerical approach, verified in this paper, is called the diffuse approximate method. The method is formed by using the weighted least squares approximation, where the second-order polynomial basis and Gaussians are used as trial and weight functions, respectively. The method is localised with the use of subdomains, each containing thirteen computational nodes. The explicit Euler scheme is used to perform the temporal integration. The fractional step method is used to couple the pressure-velocity fields. The stability of the method is attained by an adaptive shift of the computational node and Gaussian weight in the upstream direction. Results are presented for three geometrically different simulations. The results are compared with the classical finite volume method. All results show a very good agreement with the finite volume method. The simulations are performed on a uniform equidistant node arrangements of different node densities and the convergence of the node spacing is evaluated and compared. The results can also be used as a benchmark for other numerical methods.